As a tornado moves, its speed increases. The functions S(d) = 93logd+65 relates the speed of the wind, S, in miles per hour, near the center of a tornado to the distance that the tornado has traveled,d, in miles.
a) Graph this function.
b) Calculate the average rate of change for the speed of the wind at the center of a tornado from mile 10 to mile 100.
c) Estimate the rate at which the speed of the wind at the center of a tornado is changing at the moment it has traveled its 10th mile and its 100th mile.
d) Use your graph to the discuss how the rate at which the speed of the wind at the center of a tornado changes as the distance that the tornado travels increases.
http://www.wolframalpha.com/input/?i=93log(d)%2B65+for+d%3E0
Note that it's just the usual log graph, shifted up and scaled.
a) To graph the function S(d) = 93logd + 65, first plot several points and connect them with a smooth curve. Here are a few sample points:
d = 1: S(1) = 93log(1) + 65 ≈ 65
d = 10: S(10) = 93log(10) + 65 ≈ 109
d = 50: S(50) = 93log(50) + 65 ≈ 186
d = 100: S(100) = 93log(100) + 65 ≈ 221
Plot these points and any others you choose and draw a smooth curve passing through them.
b) The average rate of change for the speed of the wind from mile 10 to mile 100 can be calculated using the formula:
Average rate of change = (S(100) - S(10))/(100 - 10)
Substituting the values from part a):
Average rate of change = (221 - 109)/(100 - 10) = 112/90 ≈ 1.244
So, the average rate of change for the speed of the wind at the center of a tornado from mile 10 to mile 100 is approximately 1.244 miles per hour per mile.
c) To estimate the rate at which the speed of the wind at the center of a tornado is changing at the moment it has traveled its 10th mile and its 100th mile, we can calculate the derivative of the function S(d) with respect to d and evaluate it at those points.
The derivative of S(d) = 93logd + 65 is:
S'(d) = 93(1/d)
Evaluating at d = 10:
S'(10) = 93(1/10) = 9.3
Evaluating at d = 100:
S'(100) = 93(1/100) = 0.93
Therefore, the rate at which the speed of the wind at the center of a tornado is changing at the 10th mile is approximately 9.3 mph per mile, and at the 100th mile is approximately 0.93 mph per mile.
d) Based on the graph, we can observe that the rate at which the speed of the wind at the center of a tornado changes as the distance that the tornado travels increases is decreasing. Initially, as the distance increases, the rate of change is relatively higher, but as the distance continues to increase, the rate of change gradually decreases, approaching zero.
a) To graph the function S(d) = 93logd+65, we can first identify the key characteristics of the graph. Since logd is the input, it is important to note that d must be greater than 0. Additionally, the coefficient 93 affects the vertical stretch, and the constant 65 shifts the graph vertically. Let's plot the graph by choosing some values for d and calculating the corresponding values for S:
- Choose a few values for d, such as d = 1, 10, 100, 1000.
- Calculate S for each value: S(1) = 93log1+65 = 65, S(10) = 93log10+65 = 149, S(100) = 93log100+65 = 227, S(1000) = 93log1000+65 = 305.
Plot these points on a graph and connect them with a smooth curve. Remember to label the axes and add any necessary units.
b) To calculate the average rate of change for the speed of the wind from mile 10 to mile 100, we need to find the change in S over the change in d.
Average Rate of Change (ARC) = (S(100) - S(10))/(100-10)
Using the values we calculated earlier, the average rate of change is given by (227 - 149)/(100-10) = 78/90 = 13/15 miles per hour per mile.
c) To estimate the rate at which the speed of the wind at the center of a tornado is changing at the moment it has traveled its 10th mile and its 100th mile, we can calculate the derivatives at those specific points.
The derivative of the function S(d) = 93logd+65 represents the rate of change at any point on the graph. So, we need to find the derivatives at d = 10 and d = 100.
S'(d) = 93(1/d) (Using the derivative of the logarithm function)
S'(10) = 93(1/10) = 9.3 miles per hour per mile at mile 10.
S'(100) = 93(1/100) = 0.93 miles per hour per mile at mile 100.
d) From the graph, we can observe that as the distance that the tornado travels increases, the rate at which the speed of the wind at the center of the tornado changes decreases. Initially, the graph has a steep slope, indicating a rapid increase in wind speed. However, as the distance increases, the slope gradually decreases, indicating a slower increase in wind speed. This suggests that the rate of change is slowing down.